Finite Group Databasebeta

← C2×#26041716 #37546845

Subgroups — C2×#26041716 #37546845

Order 23168.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19153839C211yesmaximal, characteristic, complemented
2C2×D8×D181C211yesmaximal, characteristic, complemented
3#26041716C218yesmaximal, direct product factor
4C2×#19153759C211yesmaximal, characteristic, complemented
5C2×D2896C211yesmaximal, characteristic, complemented
6C2×#19153757C211yesmaximal, characteristic, complemented
7C2×#19153755C211yesmaximal, characteristic, complemented
8C2×SD32×C181C211yesmaximal, characteristic, complemented
9C2×#15668450—21no—
10C2×D4×D181—21no—
11#19153839—22no—
12#19153839C2214yescomplemented
13C2×#15668436—21no—
14C2×D1448C2211yescharacteristic
15C2×D1448C2211yescharacteristic, complemented
16C2×#15668434—21no—
17D8×D181—22no—
18D8×D181C2214yescomplemented
19C2×C8×D181C2211yescharacteristic
20#19153759C2214yescomplemented
21D2896C2214yescomplemented
22#19153757C2214yescomplemented
23#19153755C2214yescomplemented
24C2×#15668424C2211yescharacteristic, complemented
25C2×Q16×C181C2211yescharacteristic, complemented
26C2×D8×C181C2211yescharacteristic
27SD32×C181C2214yescomplemented
28C2×C16×C181C2211yescharacteristic, complemented
29C23×D181—41no—
30#15668450—24no—
31#15668450—42no—
32C2×D724—21no—
33C2×D724—21no—
34C2×D724—41no—
35C2×C22⋊D181—41no—
36D4×D181—24no—
37D4×D181—42no—
38C2×C4×D181—41no—
39C2×C4×D181D411yescharacteristic
40#15668436—24no—
41D1448—21no—
42D1448—21no—
43D1448C2312yes—
44D1448C2312yescomplemented
45#15668434—24no—
46C2×#13089678D411yescharacteristic
47C8×D181D412yes—
48C8×D181C2312yes—
49#15668424C2312yescomplemented
50C2×Q8×C181—21no—
51C2×D4×C181—21no—
52Q16×C181—21no—
53Q16×C181C2312yescomplemented
54D8×C181—21no—
55D8×C181C2312yes—
56C2×C8×C181C2311yescharacteristic
57C16×C181C2312yescomplemented
58C22×D181—41no—
59C22×D181—41no—
60C22×D181—44no—
61C22×D181—84no—
62C22×D181D811yescharacteristic
63D724—22no—
64D724—22no—
65D724—44no—
66D724—41no—
67D724—41no—
68C22⋊D181—44no—
69C2×Dic181D811yescharacteristic
70C4×D181—44no—
71C4×D181D812yes—
72C4×D181C2×D412yes—
73#13089678C2×D412yes—
74C23×C181—41no—
75Q8×C181—22no—
76Q8×C181—41no—
77D4×C181—22no—
78D4×C181—41no—
79C2×C4×C181—41no—
80C2×C4×C181C2×D411yescharacteristic
81C8×C181C2×D411yescharacteristic
82C8×C181C2411yescharacteristic
83C22×C181—42no—
84C22×C181—82no—
85C22×C181C2×D811yescharacteristic
86C2×D181—22no—
87C2×D181—42no—
88C2×D181—42no—
89C2×D181—88no—
90C2×D181—82no—
91C2×D181—82no—
92C2×D181C2×D812yes—
93C4×C181—42no—
94C4×C181C2×D811yescharacteristic
95C4×C181C22×D411yescharacteristic
96Dic181C2×D812yes—
97C2×C181—82no—
98C2×C181#50912yescomplemented
99C2×C181C22×D811yescharacteristic
100D181—22no—
101D181—82no—
102D181—82no—
103C181C2×#50911yescharacteristic, complemented, Sylow
104C2×#509—1811nomaximal, Sylow
105C2×#135—1811no—
106C22×D8—1811no—
107#509—1818no—
108C2×SD32—1811no—
109C2×SD32D18111yescharacteristic, complemented
110C2×D16—1811no—
111C2×D16—1811no—
112C2×M32—1811no—
113C2×#41—3621no—
114C22×D4—3621no—
115#135—3622no—
116#135—1814no—
117C2×Q16C2×D18111yescharacteristic, complemented
118C2×SD16—3621no—
119C2×D8—3622no—
120C2×D8—3621no—
121C2×D8—1814no—
122C2×D8—1811no—
123C2×D8—1811no—
124C2×D8C2×D18111yescharacteristic
125C22×C8—1811no—
126SD32—1814no—
127SD32C2×D18114yescomplemented
128D16—1814no—
129D16—1814no—
130M32—1814no—
131C2×C16—1811no—
132C2×C16C2×D18111yescharacteristic, complemented
133C24—7241no—
134#41—3624no—
135#41—7242no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—3624no—
139C2×D4—3621no—
140C2×D4—3621no—
141C2×D4—7241no—
142C2×D4—7242no—
143C2×D4—7241no—
144C22×C4—1811no—
145C22×C4—7241no—
146Q16—21no—
147Q16C22×D18112yescomplemented
148SD16—3624no—
149D8—21no—
150D8—3624no—
151D8—3621no—
152D8—3621no—
153D8—1812no—
154D8—1812no—
155D8C22×D18112yes—
156C2×C8—1811no—
157C2×C8—1812no—
158C2×C8—1812no—
159C2×C8C22×D18111yescharacteristic
160C16—1812no—
161C16C22×D18112yescomplemented
162C23—41no—
163C23—14484no—
164C23—1811no—
165C23—7244no—
166C23—7241no—
167C23—7241no—
168Q8—22no—
169Q8—41no—
170D4—22no—
171D4—41no—
172D4—3622no—
173D4—3622no—
174D4—7244no—
175D4—7244no—
176D4—7241no—
177D4—7241no—
178C2×C4—41no—
179C2×C4—1811no—
180C2×C4—1812no—
181C2×C4—1812no—
182C2×C4—7244no—
183C2×C4D4×D18111yescharacteristic
184C8—1812no—
185C8D4×D18111yescharacteristic
186C8C23×D18111yescharacteristic
187C22—42no—
188C22—82no—
189C22—3622no—
190C22—14488no—
191C22—14482no—
192C22—14482no—
193C22—1812no—
194C22—7242no—
195C22—7242no—
196C22D8×D18111yescharacteristic
197C4—42no—
198C4—1812no—
199C4D8×D18111yescharacteristic
200C4C2×D4×D18111yescharacteristic
201C2—82no—
202C2—3622no—
203C2—14482no—
204C2—14482no—
205C2#2604171612yesdirect product factor
206C2C2×D8×D18111yescharacteristic
subgroupG/H
centerC22D8×D181
commutator, cosocleC8×C181C24
FrattiniC8C23×D181
FittingC2×SD32×C181C2
socleC22×C181C2×D8
hypercenterC2×SD32D181
hypocenterC181C2×#509
LCSG ▷ C8×C181 ▷ C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×SD32

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#19153839C2semidirect
2. C2×D8×D181C2semidirect
3. #26041716C2direct (N × H)
4. C2×#19153759C2semidirect
5. C2×D2896C2semidirect
6. C2×#19153757C2semidirect
7. C2×#19153755C2semidirect
8. C2×SD32×C181C2semidirect
12. #19153839C22semidirect
15. C2×D1448C22semidirect
18. D8×D181C22semidirect
20. #19153759C22semidirect
21. D2896C22semidirect
22. #19153757C22semidirect
23. #19153755C22semidirect
24. C2×#15668424C22semidirect
25. C2×Q16×C181C22semidirect
27. SD32×C181C22semidirect
28. C2×C16×C181C22semidirect
44. D1448C23semidirect
49. #15668424C23semidirect
53. Q16×C181C23semidirect
57. C16×C181C23semidirect
98. C2×C181#509semidirect
103. C181C2×#509semidirect
109. C2×SD32D181semidirect
117. C2×Q16C2×D181semidirect
127. SD32C2×D181semidirect
132. C2×C16C2×D181semidirect
147. Q16C22×D181semidirect
161. C16C22×D181semidirect
205. C2#26041716direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#191538392
C2×D8×D1812
#260417162
C2×#191537592
C2×D28962
C2×#191537572
C2×#191537552
C2×SD32×C1812
C2×#509181